This study introduces a taxonomy of stereotype content in contemporary large language models (LLMs). We prompt ChatGPT 3.5, Llama 3, and Mixtral 8x7B, three powerful and widely used LLMs, for the characteristics associated with 87 social categories (e.g., gender, race, occupations). We identify 14 stereotype dimensions (e.g., Morality, Ability, Health, Beliefs, Emotions), accounting for ~90% of LLM stereotype associations. Warmth and Competence facets were the most frequent content, but all other dimensions were significantly prevalent. Stereotypes were more positive in LLMs (vs. humans), but there was significant variability across categories and dimensions. Finally, the taxonomy predicted the LLMs' internal evaluations of social categories (e.g., how positively/negatively the categories were represented), supporting the relevance of a multidimensional taxonomy for characterizing LLM stereotypes. Our findings suggest that high-dimensional human stereotypes are reflected in LLMs and must be considered in AI auditing and debiasing to minimize unidentified harms from reliance in low-dimensional views of bias in LLMs.